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One-relator groups and algebras related to polyhedral products

Part of: Low-dimensional topology Homotopy theory Algebraic combinatorics Homology and homotopy of topological groups and related structures Special aspects of infinite or finite groups

Published online by Cambridge University Press:  15 January 2021

Jelena Grbić ,
George Simmons ,
Marina Ilyasova  and
Taras Panov Open the ORCID record for Taras Panov [Opens in a new window]
Jelena Grbić
Affiliation:
School of Mathematical Sciences, University of Southampton, UK (j.grbic@soton.ac.uk; g.j.h.simmons@soton.ac.uk)
George Simmons
Affiliation:
School of Mathematical Sciences, University of Southampton, UK (j.grbic@soton.ac.uk; g.j.h.simmons@soton.ac.uk)
Marina Ilyasova
Affiliation:
Department of Mathematics and Mechanics, Moscow State University, Russia (marina_ilyasova@bk.ru)
Taras Panov
Affiliation:
Department of Mathematics and Mechanics, Moscow State University, Moscow, Russia Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia (tpanov@mech.math.msu.su)
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Abstract

We link distinct concepts of geometric group theory and homotopy theory through underlying combinatorics. For a flag simplicial complex $K$, we specify a necessary and sufficient combinatorial condition for the commutator subgroup $RC_K'$ of a right-angled Coxeter group, viewed as the fundamental group of the real moment-angle complex $\mathcal {R}_K$, to be a one-relator group; and for the Pontryagin algebra $H_{*}(\Omega \mathcal {Z}_K)$ of the moment-angle complex to be a one-relator algebra. We also give a homological characterization of these properties. For $RC_K'$, it is given by a condition on the homology group $H_2(\mathcal {R}_K)$, whereas for $H_{*}(\Omega \mathcal {Z}_K)$ it is stated in terms of the bigrading of the homology groups of $\mathcal {Z}_K$.

Keywords

one-relator groups and algebrascommutator subgroups of right-angled Coxeter groupsmoment-angle complexesflag complexes

MSC classification

Primary: 20F55: Reflection and Coxeter groups 20F65: Geometric group theory 55P15: Classification of homotopy type
Secondary: 05E45: Combinatorial aspects of simplicial complexes 57M07: Topological methods in group theory 57T30: Bar and cobar constructions

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 1 , February 2022 , pp. 128 - 147
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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